Electrons in bilayer graphene possess an unusual property: they are chiral quasiparticles characterized by Berry phase 2pi. We review the tight-binding model of bilayer graphene which determines the band structure and low-energy quasiparticle properties of this material and we describe the optical manifestation of the existence of a pair of split-bands and low-energy branches in the bilayer spectrum. Then, we analyze the stability of a bilayer with respect to a ferroelectric transition and we model the self-consistent control of the interlayer asymmetry gap induced by a transverse electric field in a graphene-based field-effect transistor.